slide 2.1- 1 copyright © 2007 pearson education, inc. publishing as pearson addison-wesley

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

OBJECTIVES

The Coordinate Plane

Plot points in the Cartesian coordinate plane.

Find the distance between two points.

Find the midpoint of a line segment.

SECTION 2.1

1

2

3


DefinitionsAn ordered pair of real numbers is a pair of real numbers in which the order is specified, and is written by enclosing a pair of numbers in parentheses and separating them with a comma.

The ordered pair (a, b) has first component a and second component b. Two ordered pairs (x, y) and (a, b) are equal if and only if x = a and y = b.The sets of ordered pairs of real numbers are identified with points on a plane called the coordinate plane or the Cartesian plane.


DefinitionsWe begin with two coordinate lines, one horizontal (x-axis) and one vertical (y-axis), that intersect at their zero points. The point of intersection of the x-axis and y-axis is called the origin. The x-axis and y-axis are called coordinate axes, and the plane formed by them is sometimes called the xy-plane.

The axes divide the plane into four regions called quadrants, which are numbered as shown in the next slide. The points on the axes themselves do not belong to any of the quadrants.


DefinitionsThe figure shows how each ordered pair (a, b) of real numbers is associated with a unique point in the plane P, and each point in the plane is associated with a unique ordered pair of real numbers. The first component, a, is called the x-coordinate of P and the second component, b, is called the y-coordinate of P, since we have called our horizontal axis the x-axis and our vertical axis the y-axis.


DefinitionsThe x-coordinate indicates the point’s distance to the right of, left of, or on the y-axis. Similarly, the y-coordinate of a point indicates its distance above, below, or on the x-axis. The signs of the x- and y-coordinates are shown in the figure for each quadrant. We refer to the point corresponding to the ordered pair (a, b) as the graph of the ordered pair (a, b) in the coordinate system. The notation P(a, b) designates the point P in the coordinate plane whose x-coordinate is a and whose y-coordinate is b.


EXAMPLE 1 Graphing Points

Graph the following points in the xy-plane:A 3,1 , B 2, 4 , C 3, 4 , D 2, 3 , E 3,0

Solution

A 3,1 3 units right, 1 unit up

3 units left, 4 units downC 3, 4 2 units left, 4 units upB 2, 4

3 units left, 0 units up or downE 3,0 2 units right, 3 units downD 2, 3


EXAMPLE 1 Graphing Points

Solution continued


EXAMPLE 2Plotting Data on Adult Smokers in the United States

The data in table below show the prevalence of smoking among adults aged 18 years and older in the United States over the years 1997–2003.

Plot the graph of the ordered pairs (year, %), where the first coordinate represents a year and the second coordinate represents the percent of adult smokers in that year.

Year 1997 1998 1999 2000 2001 2002 2003

% 24.7 24.1 23.5 23.2 22.7 22.4 21.6


EXAMPLE 2Plotting Data on Adult Smokers in the United States

Solution


THE DISTANCE FORMULA IN THE COORDINATE PLANE

Let P(x1, y1) and Q(x2, y2) be any two points in the coordinate plane. Then the distance between P and Q, denoted d(P,Q), is given by the distance formula:

d P,Q x2 x1 2 y2 y1 2.



d P,Q x2 x1 2 y2 y1 2


EXAMPLE 3

Find the distance between the points P(–2, 5) and Q(3, – 4).

Finding the Distance Between Two Points

x1 2, y1 5, x2 3, y2 4

SolutionLet (x1, y1) = (–2, 5) and (x2, y2) = (3, – 4).

d P,Q x2 x1 2 y2 y1 2

3 2 2 4 5 2

52 9 2

25 81

106

10.3


EXAMPLE 4

Let A(4, 3), B(1, 4) and C(–2, – 4) be three points in the plane.

a. Sketch the triangle ABC.

b. Find the length of each side of the triangle.

c. Show that ABC is a right triangle.

Identifying a Right Triangle


EXAMPLE 4

Solution


a. Sketch the triangle ABC.


EXAMPLE 4

Solution continued


d A, B 4 1 2 3 4 2 9 1 10

d B,C 1 2 2 4 5 2

9 81 90

d B,C 4 2 2 3 5

2

36 64 100 10

b. Find the length of each side of the triangle.


EXAMPLE 4

Solution continued


Check that a2 + b2 = c2 holds in this triangle, where a, b, and c denote the lengths of its sides. The longest side, AC, has length 10 units.

d A, B 2 d B,C

210 90

100 10 2 d A,C 2.

It follows from the converse of the Pythagorean Theorem that the triangle ABC is a right triangle.

c. Show that ABC is a right triangle.


EXAMPLE 5

The baseball “diamond” is in fact a square with a distance of 90 feet between each of the consecutive bases. Use an appropriate coordinate system to calculate the distance the ball will travel when the third baseman throws it from third base to first base.

Applying the Distance Formula to Baseball

Solution

We can conveniently choose home plate as the origin and place the x-axis along the line from home plate to first base and the y-axis along


EXAMPLE 5

Solution continued


the line from home plate to third base. The coordinates of home plate (O), first base (A) second base (C) and third base (B) are shown.


EXAMPLE 5

Find the distance between points A and B.

Solution continued


d A, B 90 0 2 0 90 2

90 2 90 2

2 90 2

90 2

127.28 feet


THE MIDPOINT FORMULA

The coordinates of the midpoint M(x, y) of the line segment joining P(x1, y1) and Q(x2, y2) are given by

x, y x1 x2

2,y1 y2

2

.



x, y x1 x2

2,y1 y2

2


EXAMPLE 6

Find the midpoint of the line segment joining the points P(–3, 6) and Q(1, 4).

Finding the Distance Between Two Points

x1 3, y1 6, x2 1, y2 4

SolutionLet (x1, y1) = (–3, 6) and (x2, y2) = (1, 4).

Midpoint x1 x2

2,y1 y2

2

31

2,6 4

2

1, 5

slide 2.1- 1 copyright © 2007 pearson education, inc. publishing as pearson addison-wesley

Documents

cartesian coordinate

coordinate axes

coordinate lines

coordinate planeexample

coordinate system

points distance

plane p

definitionsthe xcoordinate